Given a square (graphed on the Cartesian coordinate system) and a point
in the square, draw a line through the point that will divide the square
into two regions: one the smallest area possible, the other the largest
possible.

Suppose that a square billard table has corners at (0,0),(1,0),(1,1),
(0,1). A ball leaves the origin along a line with slope s. If the ball
reaches a corner, it stops or falls in. Whenever it hits the side of
the table not at a corner, it continues to travel on the table, but
the slope on the line is multiplied by -1. Does the ball reach a corner?

A student struggles to calculate matrices that correspond to linear transformations that
occur off of the x- or y-axes. Doctor Schwa formalizes what the student has already done, then introduces a second approach and the concept of conjugations.

Seeking the circumcenter of a triangle drawn on the coordinate plane, a student
determines the midpoints and slopes of all three legs -- but the equation for only one
of its perpendicular bisectors. Doctor Schwa offers a small hint that she uses to finish
off the proof.